The Main Properties And Applications Of The Compound Poisson-Geometric Process In Insurance

We often define the classical risk model as follows: it supposes that a insurance company have certain initial capital, the insurance company can accept the risk insurance with some statistical distribution restricted within law and regulation. According to the trait of the risk , the insurance company may collects premium in succession , it often takes the time as a continuous variable. In the past, many researchers have studied the continuous risk model with compound poisson process, it is also be called classical risk model or Cramér-Lundberg risk model.We know that poisson distribution has a important trait that it's mathematical expectation is the same with variance, but in practice, the claim numbers in insurance may not exactly comply with the poisson distribution's regulations. We can find that it's mathematical expectation is often greater than variance, comparatively speaking, this phenomenon is called over-dispersion. In insurance practice, the reason of over-dispersion may be some aspects as follows: the first, influenced by the natural environment and the objective condition, the accident number of individual insurance policy may stray from poisson distribution ;the second, the insurance company and the policy holder strengthened the risk consciousness, so the probability of 0 accident become greater; the third, the insurance company adopt some measures to evade risk, such as franchise and no claim discount, under these measures , the policy holder may weigh the benefit and then decide whether to claim, so the accident number may greater than the claim number. So we must deduce the poisson risk model, then it can well describe the survival probability, ruin probability, the ruin time and the greatest surplus distribution of the insurance company.In this paper, the Compound Poisson-Geometric process has be introduced as a generalization of Poisson process, the Compound Poisson-Geometric process comply with the insurance well, researched the main properties of Compound Poisson-Geometric process and illustrated the simple applications of the process. First, deduced the classical risk model under the policy of franchise, attained the process's eigenfunction,moment generating function,mathematical expectation and variance by the mothod of c thoery and stochastic process.Second, disscussed the upper bound estimation of Compound Poisson-Geometric process's ruin probability, obtained some important inferences.Third, the paper analysed the capital distribution at the point of bankruptcy, considered under the specific franchise, the insuer how to determine the pure premium.